Intonation, a Physical Phenomenon

Not on the violin, viola, cello or bass. And the physical proof lies not just in our ears, but in the science of acoustics.

John Burton, cellist and music professor at the University of Texas at Arlington, explained how the physics of sound relates to good tone in a lecture about "Resonance, Intonation, Tone: The Secret to Playing In Tune" at the 2015 ASTA conference in March. Introducing the subject, he asked how many in the audience of about 30 music teachers had ever taken a class in acoustics, and only three in the room raised their hands. (I was not among them!)

Though string players tend not to have formal education on the subject, the science of acoustics lies at the heart of what we do and how we do it.

"When you are playing the violin, viola, cello or bass, you're playing a complex standing wave," Burton said. "We have the ability to be very discriminant, when it comes to intonation."

Playing with good tone involves producing sound waves that resonate with the instrument and its strings -- it involves precision of pitch and just the right amount of force and motion with the bow.

What exactly happens, when a string player bows a string? In terms of physics: when the string is still, it is in equilibrium. The force of friction from the bow disrupts that equilibrium and makes the string move. Watching a down-bow in super-slow-motion: the bow moves the string a tiny bit to the right, then the "restoring forces" of the string make it break free of the bow and snap back to the left. These happens over and over, as the bow moves across the string, constantly grabbing and releasing it along the way.

The string resists the force of the bow; it wants to be in equilibrium. If you were to simply pluck a string once, it would vibrate but then return to equilibrium. The bow, by contrast, sets up a continuously oscillating system, whereby the string is "plucked" hundreds of tiny times and kept in vibration by applying that friction continuously from the bow.

The length of the string determines the pitch at which it vibrates. When we put down a finger to make a note, this in effect shortens the string to change the pitch.

The vibration from one string can set into motion other vibrations, and this is called "resonance." It happens on our instruments when, say, you play the note "G" -- third finger on the D string. When played perfectly in tune, the vibration of that "G" will also set the "G" string vibrating. In fact, it could also set anything in the room that is tuned to a G -- a string inside a piano, a string on the mandolin on the wall -- vibrating. If you hit a tuning fork, and another tuning fork set to the same pitch is sitting across the room, it will likely vibrate in sympathy, or "resonate."

"Anything tuned to that pitch should vibrate," Barton said.

But consider this: the "complex standing wave" that is a string produces many frequencies, not just that "G." The "G" is called the "fundamental frequency," and that's primarily what we hear. But because of all those vibrations, many other notes are present, and these are called "overtones," or the "harmonic series." The overtones are not quite as audible, but they can be magnified if they resonate well with another string or with the wood of the instrument (or even that pitchfork across the room).

Each note has its own set of overtones, and the overtones for any note follow a pattern, based on physics. The overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's "fundamental" wavelength. For us musicians, those fractions represent notes that are certain intervals above that "fundamental" note. They always follow a pattern (look at this chart from bottom to top):

So in this case, the G would be called the "fundamental" wavelength, and the overtones are portions of that wavelength and would produce the additional frequencies at higher octaves: G, D, G, B, D, F, G and it goes on.

What's remarkable, and what I did not know, is that you can see this phenomenon! With our instruments right under our chins and with the vibrations so high and small, we don't have as much opportunity to watch, but on a cello, the demonstration is a real revelation. So for the cello, let's talk about the note "C." Here is the "harmonic series," or the overtones, that are produced by the note "C", this time written as music:

"When I play a 'C,' those harmonics are present in the sound," Burton said. We mostly hear the fundamental (the C), but we also hear a lot of other vibrations: The "C" that is an octave above, the "G" a fifth above that, the next "C" a perfect fourth up, an "E" a third above that, and this continues for some 16 fractions of wavelengths and beyond. Those overtones can be magnified if they resonate with another string.

Barton showed that when he bows the "C" string on the cello, the overtones cause the "G" string to resonate, because G is an overtone (the "third partial") of C. Looking up close, one can see the "C" string vibrate, and also, one sees that the "G" vibrates. Interestingly, it vibrates in a way that looks a lot like the third example up on the wave chart: at two points of amplitude. Since I was sitting at the front of the class, I got to go up close and look at the string, vibrating in those two places. Physics in motion, check it out!

So not only can you make another string vibrate sympathetically with a specific note, but any of that note's overtones can also set a string vibrating.

This science demonstration has some implications for what we call "good tone" on our stringed instruments. Basically, "we're trying to create resonance on our instruments," Burton said. The best players make their instruments resonate as much as possible. And science shows us that our instruments resonate when we create pitch in a way that gets the overtones to ring.

"Resonance happens in two directions," Burton said. "If I play a fundamental, the notes that are predisposed to vibrate at that frequency will vibrate." On a cello, playing the note "D" on the C-string will cause both the D and A strings to vibrate, because those notes are overtones (the second and third partials) of that D.

It also happens in reverse: if you play a note that is an overtone, you can make the fundamental resonate. For example on the violin, a well-played "D" on the A-string might cause the G-string to resonate because that "D" is an overtone of "G."

Of course, this does not happen with every note on the instrument.

"All notes are good on cello, but some notes are gooder than others!" Burton joked. Not every pitch will have a harmonic series that relates to the instrument's strings.

Burton pointed out some really cool stuff on the cello, like: If you play a high G, then the G string will vibrate in four equal lengths, something you can feel better than you can see. Also, the vibrations on the G string have two points of amplitude, when you play the "G" that is one octave higher. You can also finger a note to make the string vibrate in sympathy with another note on another string.

Though these phenomenon are integral to the violin as well, they aren't as easy to demonstrate because "the shorter the string, the greater the tension, and the harder it is to see," Burton said.

Suffice it to say this: "intonation and tone are synonymous," Burton said. When you play a note that is even slightly out-of-tune, "it's dead, it has no ring," he said. An out-of-tune pitch will not set any of those resonances in motion. If one plays pitches, with no awareness or feeling for resonance, it's very hard to find the voice of the instrument. "The more I can drill the sound of this cello, associated with the resonance of these notes, the more I can teach pitch."

Laurie, thanks so much for this informative post. I think many violinists already understand this "resonance" idea, but having it spelled out so clearly is a valuable resource. I agree that a course in acoustics would be useful for musicians but I don't think one needs that course to understand what you have described here.

Another general principle is that the fifths on the violin are wider than the fifths on the piano (equal-tempered fifths) by about 0.3%. Thus notes that are referenced by "resonance" to another string will also deviate from the corresponding note in equal temperament.

Look at the picture that is called "Harmonic_series.jpg" with the notes numbered 1-16. The notes that are marked "2", "4", and "8" are octave harmonics. These notes will be just as "in tune" as the open G, there is no additional deviation from equal temperament (percentage-wise). For the note marked "3" the additional deviation is very small. However, for the note marked "5" and especially the note marked "7" the deviations are fairly large. I did not go beyond 8 (there's not that much point, as the issues become redundant). Some may feel that I'm setting up something of a straw man by introducing equal temperament as a reference point, but when a particular note (or two) on the violin deviates from equal temperament by a large amount, then I think that's at least worth discussing.

The note marked "5" corresponds, on the violin, to the "B" that we would play with our fourth finger in first position on the E string. Your article suggests that notes are "in tune" by definition when they are on-resonance, and this particular resonance is quite easy to hear even though it is not always taught to students as one of the "ring tones". Play a slow G major scale and be sure to finger that "B" on the E string and make sure it rings nicely. Does it sound in tune to you? Or does it sound flat?

So my question to you is, do you teach your students that when they play that "B" on the E string, will it *always* be in tune when it is "on resonance" with the open G string, regardless of context?

From 98.118.42.5
Posted on April 13, 2015 at 6:03 PM

Paul brings up some good observations.

First let's look at the minor 7th. By sheer coincidence, the 7th harmonic happens to make a perfectly sweet consonant chord with a "7" feel to it. It it very noticeably flatter than *either* the pythagorean minor 7 or even the "just" minor 7 that falls out of making diatonics from low ratio harmonics.

This special minor 7 chord is actually used by good guitarists--when playing in open tuning (that means a chord rings without stopping the string at a fret) the chord is adjusted away from equal temperament. This is most commonly done on guitar to get the third to be sweet--but some guitarists do it with the 7th harmonic, too!

As for teaching this---that seems in my observation--to be left out unless you are studying Carnatic violin or Persian violin etc where "microtones" are important. Haha. Which brings up another point: everything has some sort of resonance. Just not the one you are used to :-)

From 98.118.42.5
Posted on April 13, 2015 at 6:11 PM

Here is John Catler demonstrating a pure A7 chord...https://www.youtube.com/watch?v=O2eukIoSsKM

All great points, I welcome people's comments on it, as I'm definitely not an expert in the physics of acoustics! It's definitely an interesting and relevant topic for us violinists, though.

From 72.190.77.11
Posted on April 13, 2015 at 8:44 PM

Laurie:

This is a most excellent blog and you should repeat it every year in April.

There is one element (maybe too elementary to mention). While sympathetic vibrations can be passed by the air, on the violin families of instruments, there is a very solid mechanical connection among all of the strings. It is the bridge.

Every pulse created by the bow on one string is transmitted by the top edge of the bridge to the other three strings. Therefore they can be uncommonly excited and produce glorious overtones when the fundamental played note is in tune.

This vibration is carried into the tailpiece which pivots by the flexibility of the tail gut on the saddle.

A great article. But intonation on a violin (and the other instruments in the violin family) can be quite a complicated matter with many other aspects; the more you learn on this matter the more aspects there seem to be.

The western music tradition has adopted equal temperament as the standard, but violinists don't stick to that depending on the circumstances like the style of the music and how you tune the violin.

Some tuning possibilities:

You can tune the violin in perfect fifths but you might get in trouble sometimes.

If you play with a piano you can tune from a G9 chord. That is on the piano play from the G that is the same pitch as the violin G-string: G-B-D-F-A. Simply tune to that chord.

Some violin players, in an orchestra, tune the A slightly high and then the other strings in perfect fifths. That will compensate a bit so the lower strings won't feel that low in tuning, but the E-string will be quite high. I don't know how common that is, but I did once meet a bassoon player that was a bit annoyed about the violin players' slightly high A.

QUOTESo my question to you is, do you teach your students that when they play that "B" on the E string, will it *always* be in tune when it is "on resonance" with the open G string, regardless of context? UNQUOTE

If you play the B a perfect fifth higher than the E-string it will be too high compared with the G-string. I think that if the key of the tune you play is E-major your ear will probably want the high version of the B. The high B actually has a great resonance. If you play in G-major the ear is more likely to want the low B. So my answer is that it won't always be in tune if it resonates with the G-string.

I teach my pupils to intonate the pitches that are octaves compared with the open strings, G-D-A-E, and resonates with them, as an important focus regarding intonation, a kind of basis. I haven't focused on tuning the high B after the G-string.

The other day I had a lesson with an advanced pupil where we discussed intonation the way Simon Fischer suggests in his book "Basics", which includes the intonation after the open string for the notes G-D-A-E.

We then touched the idea that sometines that doesn't work when you play together with others. My pupil had found out that she had to adjust the picth on a tone a little bit in order to make it sound right when playing together. That is an example demonstrating that musicians fortunately intonate by ear very well without thinking of theory. We can practice and work with theory and evolve and get better, but the ear and the general feel when you play is vital and a wonderful part of it.

I agree with Allan B. Lewis.All this is fine when one plays solo, or teaches the applied physics, but the moment one sits in a quartet or string orchestra, the intonation becomes a group challenge. I have been working hard on my intonation intuitively the way it is described in the blog and then found myself in a Baroque orchestra where I had to "give up" or negotiate on my individual pitch in a context. The coach could not answer my question raising the paradox: how can I keep my violin resonating at its best while constantly adapting my pitch with the others?

Bring vibrato into the picture, and see what happens. Vibrato generates new frequencies either side of the base frequency. Some, at least, of the new frequencies and their harmonics may resonate with other resonances of the instrument.Further, the nature of vibrato (its rate and amplitude) means that it naturally generates other frequencies - known as sideband frequencies - which may also add to the richness of the frequency mix the listener hears. Sideband frequencies exist with any oscillating frequency source; their existence can be proved mathematically.

A timely post for me. I recently acquired a Yann Poulain violin and after a few months I realized one day that my unplayed strings were producing faint, high-pitched sounds. This helps explain why that is happening (I just thought it was because I had an excellent instrument).

As far as the sympathy vibrations, this is something that I also noticed years ago and I use as a reference point most often when playing fourth finger in first position or second in third; i.e., looking for the 'A' to vibrate when playing the note on the 'D' string, etc.

I don't know about other violinists but I can clearly hear and feel when my intonation is correct. The sound is fuller and the violin reacts to it by vibrating a lot more.